Suppose it is an unknown and a quadratic equation, and its two roots have the following relationship:
,
Both sum and have this property: if sum is exchanged, the result remains the same, because
,
Polynomials with this sum are called symmetric polynomials.
For example, it is also a symmetric polynomial, but it is not. And we are used to calling sum an elementary symmetric polynomial.
Let's look at the general situation. Let n ∈ z+, A0, A 1, ... an ∈ c, A0 ≠ 0 suppose there is a polynomial equation with a variable of degree n:
The famous basic theorem of algebra tells us that such an equation has n roots. Suppose:
Similar to the second case, Vieta theorem gives:
Like the ones on the left above:
Polynomials like this will not change no matter how we arrange them. In other words, if we arrange them into n, then the above formula will not change. Such a formula is called a symmetric polynomial, and the symmetric polynomial above is an elementary symmetric polynomial.
Definition 6: Let it be an n-ary polynomial on C, and if n in the exponent set {1, 2, ... remains unchanged after any permutation, it is called an n-ary symmetric polynomial on C. 。
For example, it is a symmetric polynomial,
It's not,
If:1→ 2,2 → 3,3 →1
therefore
The importance of elementary symmetric polynomials lies in
Theorem (the basic theorem of symmetric polynomials);
Every n-ary symmetric polynomial can be uniquely expressed as a polynomial of an elementary symmetric polynomial.
Now we use group language to describe the symmetry of n-ary polynomials.
Let Sn be the transformation group of m, that is, the n-degree symmetric group mentioned above. If we omit letters and just write down labels, the elements in Sn can be recorded as:
This is an n-shaped arrangement.
Let f count all n-ary polynomials over field F. Yes, you can define the mapping from f to f by using.
Then it is a one-to-one transformation of the set F. Why?
Make Tn enter
Then (Tn, O) is satisfied, which is called the permutation group of F. 。
If the polynomial of n variables is compared to a plane figure and F is compared to a plane, then the permutation group of F is equivalent to the motion group of the plane (all distance-preserving transformations of the plane).
That is, all invariants, then we satisfy the property and call it n-ary polynomial symmetric group.
For example 1:, then the quartic symmetric group is symmetric.
Example 2:
Example 3:
-Klein 4 yuan Group
Example 4: Unit Element Group
Example 5:
It is a third-order cyclic solution.
Definition: The polynomial of is called symmetric polynomial, if the symmetric group is the whole permutation group.
In this way, we use groups to describe the symmetry of polynomials.
How to construct symmetric polynomials can be found in modern algebra P55.
Fourthly, the symmetry of number field.
The concept of number field was mentioned in advanced algebra of freshmen.
A set of non-empty numbers f contains at least one non-zero number. If f is close to+,-,×, ⊙, then f is called a number field.
Q, r and c are all number fields, and the smallest number field is q,
It is also a numeric field.
Plane figure is a geometric structure, that is, the distance between point set M (figure is composed of points) and any two points in point set M is regarded as a whole, and its symmetric group is the complete set of M's transformations that keep the distance between any two points constant. These transformations that preserve the geometric structure (i.e. distance) of M describe the symmetry of the geometric structure.
Similarly, the number field F is an algebraic structure, that is, a number set F and the addition, subtraction, multiplication and division operations in this number set F are regarded as a whole.
Therefore, the symmetry of number field F can also be described by the integral transformation (i.e. operation) of F-preserving algebraic structure.
Defining the automorphism of 7- number field f means:
(1) is a one-to-one transformation of f.
(2)
Theorem 1 If it is an automorphism of f, it has the following properties:
( 1)
(2) ;
(3)
(4) .
Like the symmetry of plane finite graph K we discussed before, the product of two symmetric transformations is still the symmetric transformation of K. Similarly, we have:
Property 1 Let sum be two automorphisms of number field F, then sum is also an automorphism of F. 。
Property 2: Let Aut(F) represent all automorphisms of f, and let o represent multiplication of transformation, then (Aut(F), o) satisfies g1)-g4).
8-sign (Aut(F), o) is defined as an automorphism group of number field F.
We can make an analogy: the automorphism group of number field F is equivalent to the symmetry group of graph K, the latter describes the symmetry of graph K, and the former describes the "symmetry" of number field, which is an analogy concept of graph symmetry in number field.
Theorem 2 The automorphism group of rational number field has only one element-unit automorphism i. 。
Therefore, if any number field f, f, and, then, that is, the restriction is an identity transformation.
Example 1 order is a number field, which is an algebraic extension field made by addition. The automorphism group of F is studied.
set up
,
According to the theorem 1,
Therefore, the result of the conversion depends on
Let there be at most two numerical sums, so the automorphism group of f is only.
It can be proved that I and I are indeed automorphisms on F.
o I φ
I I φ
φ φ I
This is a 2 yuan cyclic group,
Symmetric groups isomorphic to, that is,.
Example 2 Order
This is also a numeric field. Suppose, as in the previous example, its function depends on four combinations of sum, knowledge and sum. Therefore, Aut(E) has only four elements.
o I φ 1 φ2 φ 12
I I φ 1 φ2 φ 12
φ 1 φ 1 I φ 12 φ2
φ2 φ2 φ 12 I φ 1
φ 12 φ 12 φ2 φ 1 I
o( 1)( 12)(34)( 12)(34)
( 1) ( 1) ( 12) (34) ( 12)(34)
( 12) ( 12) ( 1) ( 12)(34) (34)
(34) (34) ( 12)(34) ( 1) ( 12)
( 12)(34) ( 12)(34) (34) ( 12) ( 1)
Aut(E) and Klein 4 yuan group are isomorphic;
,
That is, symmetric groups.
Let's generalize the above to the general situation.
Definition 9 gives two number fields f and e. If F E and f are called subdomains of e, e is called extension fields of f..
That is, the automorphism of e that fixes the elements in f, Aut(E:F) is composed of all these.
F is equivalent to the symmetry axis or the center of rotation in the symmetry of a plane figure.
The proposition (Aut(E:F), o) is satisfied, which is called the symmetric group of number field e over f 。
Example 3
Neither and can keep a+b unchanged.
Let, be a polynomial of degree n, have n roots, and the splitting field on F is E, then (Aut(E:F), o) is called the symmetric group of the roots of polynomials on F, also called the Galois group of univariate polynomials on F, which plays a key role in solving the problem that polynomial equations of degree five or more cannot have root solutions.
Teaching of verb (Verb's abbreviation) "Symmetry and Group"
(1) To understand the universality of operation, not only numbers can be operated, but also some other mathematical objects can be operated, and it satisfies the properties of some digital operations.
(2) Multiplication is not necessarily commutative.
(3) The concept of algebraic structure: a set, plus the operations in this set, constitutes an algebraic system, and its structure is in the operational relationship.
(4) The concept of group: A symmetric group is a specific group, which satisfies G 1)-G4) and is called a group.
(5) Mathematical language is an excellent tool to describe natural phenomena, and mathematics is a research model. Mathematics comes from practical problems.